# Kuder–Richardson Formula 20

{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }} In statistics, the Kuder–Richardson Formula 20 (KR-20) first published in 1937 is a measure of internal consistency reliability for measures with dichotomous choices. It is analogous to Cronbach's α, except Cronbach's α is also used for non-dichotomous (continuous) measures. It is often claimed that a high KR-20 coefficient (e.g., > 0.90) indicates a homogeneous test. However, like Cronbach's α, homogeneity (that is, unidimensionality) is actually an assumption, not a conclusion, of reliability coefficients. It is possible, for example, to have a high KR-20 with a multidimensional scale, especially with a large number of items.

Values can range from 0.00 to 1.00 (sometimes expressed as 0 to 100), with high values indicating that the examination is likely to correlate with alternate forms (a desirable characteristic). The KR-20 may be affected by difficulty of the test, the spread in scores and the length of the examination.

In the case when scores are not tau-equivalent (for example when there is not homogeneous but rather examination items of increasing difficulty) then the KR-20 is an indication of the lower bound of internal consistency (reliability).

The formula for KR-20 for a test with K test items numbered i=1 to K is

$r={\frac {K}{K-1}}\left[1-{\frac {\sum _{i=1}^{K}p_{i}q_{i}}{\sigma _{X}^{2}}}\right]$ where pi is the proportion of correct responses to test item i, qi is the proportion of incorrect responses to test item i (so that pi + qi = 1), and the variance for the denominator is

$\sigma _{X}^{2}={\frac {\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}\,{}}{n}}.$ where n is the total sample size.

If it is important to use unbiased operators then the sum of squares should be divided by degrees of freedom (n − 1) and the probabilities are multiplied by

${\frac {n}{n-1}}$ Since Cronbach's α was published in 1951, there has been no known advantage to KR-20 over Cronbach. KR-20 is seen as a derivative of the Cronbach formula, with the advantage to Cronbach that it can handle both dichotomous and continuous variables. The KR-20 formula can't be used when multiple-choice questions involve partial credit, and it requires detailed item analysis.